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Sec. 7–2 Performance of Baseband Binary Systems 503
(
s 1 (t)
r(t)= or +n(t)
s 2 (t)
(
N 0
Low-pass filter
or
matched filter
r 0 (t)
Sample
and hold
r 0 (t 0 )=r 0
Threshold device
–V T
~ m
V T r 0
Digital
output m ~
where p n (f)= 2
Figure 7–6 Receiver for bipolar signaling.
(a) Receiver
f(r 0 |-A sent)
f
f(r 0 | s 2 A sent)
f(r 0 |+A sent)
Q (V T /Í 0 ) Q (V T /Í 0 )
–A –V T V T
±A
r 0
P (error| – A sent)=Q[(A – V T )/Í 0 ] P (error| ± A sent)=Q[(A – V T )/Í 0 ]
(b) Conditional PDFs
or
P e L Qa V T
b + 1 s 0 2 Qa A - V T
b
s 0
using calculus, we find that the optimum value of V T that gives the minimum BER is
V T = A For systems with reasonably low (i.e., usable) BERs, A 7 s 0 , so that the
2 + s 0 2
A ln2.
optimum threshold becomes approximately V T = A/2, the BER is
P e = 3 2 Qa A
2s 0
b
For the case of a receiver with a low-pass filter that has a bipolar signal plus white noise at its
input, s 2 0 = N 0 B. Thus, the BER is
A 2
P e = 3 2 Q¢ C4N 0 B ≤
(low-pass filter)
(7–28a)
where the PSD of the input noise is N 0 /2 and the equivalent bandwidth of the filter is B Hz.
If a matched filter is used, its output SNR is, using Eq. (6–161),
a S N b out
= A2
s 0
2 = 2E d
N 0